SpheroidalS1Prime

SpheroidalS1Prime[n,m,γ,z]

gives the derivative with respect to of the radial spheroidal function of the first kind.

Details

Mathematical function, suitable for both symbolic and numerical manipulation.
For certain special arguments, SpheroidalS1Prime automatically evaluates to exact values.
SpheroidalS1Prime can be evaluated to arbitrary numerical precision.
SpheroidalS1Prime automatically threads over lists.

Examples

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Basic Examples (5)

Evaluate numerically:

Plot over a subset of the reals:

Plot over a subset of the complexes:

Series expansion at the origin:

Series expansion at a singular point:

Scope (17)

Numerical Evaluation (4)

Evaluate numerically:

Evaluate to high precision:

The precision of the output tracks the precision of the input:

Complex number inputs:

Evaluate efficiently at high precision:

Specific Values (4)

Simple exact values are generated automatically:

Find the first positive maximum of SpheroidalS1Prime[2,0,5,x]:

SpheroidalS1Prime functions become elementary if m=1 and γ=n π/2 :

TraditionalForm typesetting:

Visualization (3)

Plot the SpheroidalS1Prime function for integer orders:

Plot the SpheroidalS1Prime function for noninteger parameters:

Plot the real part of :

Plot the imaginary part of :

Differentiation (2)

First derivative with respect to z:

Higher derivatives with respect to z:

Plot the higher derivatives with respect to z when n=10, m=2 and γ=1/3:

Integration (2)

Compute the indefinite integral using Integrate:

Verify the anti-derivative:

Definite integral:

Series Expansions (2)

Find the Taylor expansion using Series:

Plots of the first three approximations around :

Taylor expansion at a generic point:

Applications (1)

Find resonant frequencies for the Neumann problem in a prolate spheroidal cavity:

Determine the first few frequencies:

Possible Issues (1)

Spheroidal functions do not evaluate for half-integer values of and generic values of :

Introduced in 2007

(6.0)

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